How do you move in the direction of a euler angle?

I want to have an object move in the direction of an angle, but not actually be facing that angle.

Basic movement is simple, and looks like this:

ThisTransform.position = Vector3.MoveTowards(ThisTransform.position, _destinationPosition, MoveSpeed * Time.deltaTime);

However, this moves towards the specified point, and will stop at the point, which is not what I want. I want to continuously move in a certain stored Euler Angle. How would I achieve this?

Use movetowards but give it a negative value and swap the to and from destinations around. The value does not get clamped when using negatives. It can be used to push yourself in a direction.

Or just modify the position directly. In Update (assuming that _angle is a Vector3):
Vector3 dir = Quaternion.Euler(_angle) * Vector3.forward;
ThisTransform.position += dir * _speed * Time.deltaTime;

For better efficiency, calculate dir * _speed ahead of time.